Hamlet is preparing a play to find out the truth about his father's death. The theatre has $2015$ numbered seats and can contain all the members of King Claudius's court. He gives to each member a ticket with the number of their seat.
King Claudius (the first to sit) doesn't look at his ticket and chooses at random a seat. All the other guests arrive one by one and if their seat is available they sit according to the ticket, if it isn't they sit at random.
Ophelia is the last to enter the theatre and she occupies the only empty spot.

What's the probability that she will sit in the seat assigned to her by her ticket?

4 Answers
4

If the king sits in his own seat, then each guest will sit in their own seat and Ophelia will always sit in her own seat. This occurs with probability $1/2015$.

If the king sits in Ophelia's seat, then each guest will sit in their own seat and Ophelia will end up sitting in the king's seat. This also occurs with probability $1/2015$.

If the king sits in any other seat, then as soon as that guest arrives, he will choose another seat at random.

If the guest chooses the king's seat, then Ophelia will end up sitting in her own seat.

If the guest chooses Ophelia's seat, then Ophelia will end up sitting in the king's seat.

If the guest chooses another guest's seat, then as soon as the guest who was assigned that seat arrives, he will also have to choose one at random, and the cycle will continue until one of the above two scenarios happens.

Since the probability is equal that each guest will choose either the king's seat or Ophelia's seat, the eventual overall probability of each event is $1/2$. The above scenario happens with probability $2013/2015$.

So the grand total probability that Ophelia ends up in her own seat is $1/2015 + 0 + 1/2 \times 2013/2015 = 1/2$.

There is exactly one guest(including the king) sitting in a seat that we do not know belongs to a guest that has already arrived.

This means that when a guest arrives he will find his seat taken with probability 1/(Number of free seats+1). This continues until Ophelia arrives, so Ophelia has probability 1/(1+1) of finding her seat taken.

P(x) = 1 - [1/2015 + ΣP(1/(2015-x))] Where x is the seat # of the person who's seat was taken. Initially there is a 1 in 2015 chance the king will take Ophelia's seat. Then there is a 1/(2015-X) chance that the person who's seat the king took will take Ophelia's seat...Continue the process for all the peoples who's seats were taken, add all the probabilities up and you should get the chance her seat is taken. One minus that probability is the chance the seat is empty. The end result should be 1/2 as stated by Joe Z.